Problem: Find the greatest common factor of $18, 36,$ and $45$.
Explanation: The greatest common factor (GCF) is the largest number that is a factor of $18, 36,$ and $45$. In order to find the GCF, we can factor each number completely as a product of prime numbers: $ \begin{aligned}18 &=2\cdot3\cdot3\\\\\\\\ 36&=2\cdot2\cdot3\cdot3\\\\\\\\ 45&=3\cdot3\cdot5 \end{aligned}$ Now, let's find the factors that are common to each number: $ \begin{aligned}18 &=2\cdot3\cdot3\\\\\\\\ 36&=2\cdot2\cdot3\cdot3\\\\\\\\ 45&=3\cdot3\cdot5 \end{aligned}$ Each number shares the factors ${3}$ and $3,$ so the GCF is $3\cdot3={9}$. The greatest common factor of $18, 36,$ and $45$ is $9$.